A PROOF OF THE FABER INTERSECTION NUMBER CONJECTURE Kefeng Liu &amp

83(2009) 313-335

A PROOF OF THE FABER INTERSECTION NUMBER CONJECTURE

Kefeng Liu & Hao Xu

Abstract

We prove the famous Faber intersection number conjecture and other more general results by using a recursion formula ofn-point functions for intersection numbers on moduli spaces of curves. We also present some vanishing properties of Gromov-Witten invari- ants.

1. Introduction

Starting from the work of Mumford, one fundamental problem in algebraic geometry is the study of intersection theory on moduli spaces of stable curves. Through the work of Witten and Kontsevich we learned that the intersection theory of moduli spaces also has striking connection to string theory and two dimensional gravity. Denote by Mg,n the moduli space of stable n-pointed genusg complex algebraic curves. We have the morphism that forgets the last marked point

π:Mg,n+1 −→ Mg,n.

Denote by σ₁, . . . , σ_n the canonical sections of π, and by D₁, . . . , D_n the corresponding divisors in Mg,n+1. Let ω_π be the relative dualizing sheaf, we have the following tautological classes on moduli spaces of curves.

ψ_i =c₁(σ^∗_i(ω_π)) κ_i =π_∗

c₁

ω_πX

D_ii+1 λ_k=c_k(E), 1≤k≤g,

where E=π∗(ωπ) is the Hodge bundle.

Intuitively,ψ_iis the first Chern class of the line bundle corresponding to the cotangent space of the universal curve at the i-th marked point and the fiber ofEis the space of holomorphic one forms on the algebraic curve.

The classes κ_i were first introduced by Mumford [22] on M_g, their generalization to Mg,n here is due to Arbarello-Cornalba [1].

Received 10/29/2008.

313

We use Witten’s notation

hτd1· · ·τ_dnκ_a1· · ·κ_am |λ^k₁¹· · ·λ^k_g^gi ,

Z

Mg,n

ψ₁^d1· · ·ψ^d_nⁿκ_a1· · ·κ_amλ^k₁¹· · ·λ^k_g^g.

These intersection numbers are called the Hodge integrals. They are rational numbers because the moduli space of curves are orbifolds (with quotient singularities) except in genus zero. Their degrees should add up to dimMg,n= 3g−3 +n.

Intersection numbers of pureψclasseshτd1· · ·τ_dniare often called in- tersection indices or descendant integrals. Faber’s algorithm [3] reduces the calculation of general Hodge integrals to intersection indices, based on Mumford’s Chern character formula [22]

ch_2g−1(E) = B_2g (2g)!

"

κ_2g−1−

n

X

i=1

ψ_i^2g−1

+1 2

X

ξ∈∆

l_ξ∗

2g−2

X

i=0

ψⁱ_n+1(−ψn+2)^2g−2−i

! # , where ∆ enumerates all boundary divisors and l_ξ∗ is the push-forward map under the natural inclusion.

The celebrated Witten-Kontsevich theorem [13, 25] asserts that the generating function of intersection indices

F(t₀, t₁, . . .) =X

g

X

n

h

∞

Y

i=0

τ_iⁿⁱi_g

∞

Y

i=0

tⁿ_iⁱ n_i!

is governed by the KdV hierarchy, which provides a recursive way to compute all these intersection numbers.

The tautological ringR^∗(Mg) is defined to be the smallestQ-subalge- bra of the Chow ring A^∗(Mg) generated by the tautological classes κi

andλ_i. Mumford [22] proved that the ringR^∗(Mg) is in fact generated by the g−2 classesκ1, . . . , κg−2.

It is a theorem of Looijenga [19] that dimR^k(Mg) = 0, k > g−2 and dimR^g−2(Mg)≤1. Later Faber proved that actually dimR^g−2(Mg) = 1.

Faber’s conjecture.Around 1993, Faber [2] proposed three remark- able conjectures about the structure of the tautological ring R^∗(M_g) which we briefly state as follows:

i) For 0≤k≤g−2, the natural product

R^k(M_g)×R^g−2−k(M_g)→R^g−2(M_g)∼=Q is a perfect pairing.

ii) The [g/3] classes κ₁, . . . , κ_[g/3] generate the ringR^∗(Mg), with no relations in degrees ≤[g/3].

iii) LetPn

j=1d_j =g−2 and d_j ≥0. Then (1) π∗(ψ₁^d1+1. . . ψ_n^dn+1) = X

σ∈Sn

κσ = (2g−3 +n)!

(2g−2)!!Q_n

j=1(2d_j + 1)!!κg−2, whereκσ is defined as follows: write the permutationσ as a prod- uct of ν(σ) disjoint cycles σ =β₁· · ·β_ν(σ), where we think of the symmetric group S_n as acting on the n-tuple (d₁, . . . , d_n). De- note by |β| the sum of the elements of a cycle β. Then κ_σ = κ_|β1|κ_|β2|. . . κ_|βν(σ)|.

Part (i) is called Faber’s perfect pairing conjecture, which is still open.

Faber has verified it forg≤23.

Part (ii) has been proved independently by Morita [21] and Ionel [12] with very different methods. As pointed out by Faber [2], Harer’s stability result implies that there is no relation in degrees ≤[g/3].

Part (iii) of Faber’s conjectures is the intersection number conjecture, whose importance lies in that it computes all top intersections in the tautological ringR^∗(Mg) and determines its ring structure if we assume Faber’s perfect pairing conjecture. Theoretically it gives the dimension of tautological rings by computing the rank of intersection matrices which we will discuss in a subsequent work.

Faber’s conjecture is a fundamental question mentioned in mono- graphs such as [6,11] that many algebraic geometers have worked on.

In this paper, we prove the Faber intersection number conjecture com- pletely. First we recall two equivalent formulations.

The Faber intersection number conjecture is equivalent to (2)

Z

Mg,n

ψ₁^d1. . . ψ^d_nⁿλ_gλ_g−1= (2g−3 +n)!|B2g| 2^2g−1(2g)!Qn

j=1(2d_j −1)!!,

where B_2g denotes the 2g-th Bernoulli number. By Mumford’s for- mula for the Chern character of the Hodge bundle, the above identity is equivalent to

(3) (2g−3 +n)!

2^2g−1(2g−1)!Q_n

j=1(2d_j−1)!! = hτ2g

n

Y

j=1

τ_djig−

n

X

j=1

hτdj+2g−1

Y

i6=j

τ_diig+1 2

2g−2

X

j=0

(−1)^jhτ2g−2−jτ_j

n

Y

i=1

τ_diig−1

+ 1 2

X

n=I‘ J

2g−2

X

j=0

(−1)^jhτj

Y

i∈I

τ_diig^′hτ2g−2−j

Y

i∈J

τ_diig−g^′,

where d_j ≥1,P_n

j=1d_j =g+n−2. We refer to [2, 17] for discussions of the above equivalences.

The following interesting relation is observed by Faber and proved by Zagier using the Faber intersection number conjecture (see [2])

κ^g−2₁ = 1

g−12^2g−5((g−2)!)²κg−2.

In fact, from (1), the above relation is equivalent to a combinatorial identity

g

X

k=1





 (−1)^k

k! (2g+ 1 +k) X

g=m1+···+mk mi>0

2g+k

2m₁+ 1, . . . ,2m_k+ 1







= (−1)^g2^2g(g!)². We learned of an elegant proof from Jian Zhou using the residue theo- rem.

Faber [2] proved identity (3) whenn= 1 using explicit formulae of up to three-point functions. The identity (2) was shown to follow from the degree 0 Virasoro conjecture for P² by Getzler and Pandharipande [8].

In 2001 Givental [9] has announced a proof of Virasoro conjecture forPⁿ. Y.-P. Lee and R. Pandharipande are writing a book [16] giving details.

Recently Teleman [23] announced a proof of the Virasoro conjecture for manifolds with semi-simple quantum cohomology. His argument depends crucially on the Mumford conjecture about the stable rational cohomology rings of the moduli spaces proved by Madsen and Weiss [20].

Goulden, Jackson and Vakil [10] recently give an enlightening proof of identity (1) for up to three points. Their remarkable proof uses relative virtual localization and a combinatorialization of the Hodge integrals, establishing connections to double Hurwitz numbers.

Our alternative approach is quite direct, we prove identity (3) for all g and n by using a recursive formula of n-point functions. Actually, the n-point function formula has far-reaching applications. Recently Zhou [28] used our results on n-point functions in his computation of Hurwitz-Hodge integrals, which leads to a proof of the crepant resolution conjecture of type A surface singularities for all genera.

Acknowledgements. The authors would like to thank Professors Sergei Lando, Jun Li, Chiu-Chu Melissa Liu, Ravi Vakil and Jian Zhou for helpful communications.

2. The n-point functions

Definition 2.1. We call the following generating function F(x₁, . . . , x_n) =

∞

X

g=0

F_g(x₁, . . . , x_n)

=

∞

X

g=0

X

Pdj=3g−3+n

hτd1· · ·τ_dnig n

Y

j=1

x^d_j^j

then-point functions.

Consider the following “normalized” n-point function G(x1, . . . , xn) = exp −P_n

j=1x³_j 24

!

F(x1, . . . , xn).

We will let G_g(x₁, . . . , x_n) denote the degree 3g−3 +n homogenous component ofG(x₁, . . . , x_n).

In contrast with the original n-point function, its normalization has some distinct properties (see [18]). For example, the coefficient of z^kQn

j=1x^d_j^j inG_g(z, x₁, . . . , x_n) is zero wheneverk >2g−2 +n.

It’s well-known that

F₀(x₁, . . . , x_n) =G₀(x₁, . . . , x_n) = (x₁+· · ·+x_n)ⁿ⁻³.

There are explicit formulae for one and two-point functions due to Witten [25] and Dijkgraaf (see [2]) respectively

G(x) = 1

x², G(x, y) = 1 x+y

X

k≥0

k!

(2k+ 1)!

1

2xy(x+y) k

. In an unpublished note [27] (kindly sent to us by Faber), Zagier obtained a marvelous formula of the three-point function (see [18]).

We proved in [18] the following recursion formula for general normal- ized n-point function.

Proposition 2.2. [18]For n≥2, G(x₁, . . . , x_n) = X

r,s≥0

(2r+n−3)!!

4^s(2r+ 2s+n−1)!!P_r(x₁, . . . , x_n)∆(x₁, . . . , x_n)^s, where P_r and∆ are homogeneous symmetric polynomials defined by

∆(x₁, . . . , x_n) = P_n

j=1x_j3

−P_n

j=1x³_j

3 ,

P_r(x₁, . . . , x_n) =



 1 2P_n

j=1x_j X

n=I‘ J

X

i∈I

xi

2 X

i∈J

xi

2

G(xI)G(xJ)





3r+n−3

= 1

2Pn j=1xj

X

n=I‘ J

X

i∈I

x_i2 X

i∈J

x_i2 r

X

r^′=0

G_r^′(x_I)G_r−r^′(x_J), where I, J 6= ∅, n = {1,2, . . . , n} and G_g(x_I) denotes the degree 3g+

|I| − 3 homogeneous component of the normalized |I|-point function G(x_k1, . . . , x_k|I|), where k_j ∈I.

The proof amounts to check thatG(x₁, . . . , x_n), as recursively defined in Proposition 2.2, satisfies the following Witten-Kontsevich differential equation (see [18]),

y ∂

∂y





y+

n

X

j=1

x_j2

G_g(y, x₁, . . . , x_n)



= y

8

y+

n

X

j=1

x_j4

G_g−1(y, x₁, . . . , x_n)−y³ 8

y+

n

X

j=1

x_j2

G_g−1(y, x₁, . . . , x_n)

+y 2

X

n=I‘ J



 y+X

i∈I

x_i

! X

i∈J

x_i

!3

+ 2 y+X

i∈I

x_i

!2

X

i∈J

x_i

!2



×G_g^′(y, x_I)G_g−g^′(x_J)

−1 2



y+

n

X

j=1

x_j









n

X

j=1

x_j



G_g(y, x₁, . . . , x_n).

The verification is tedious but straightforward. It will be included in a updated version of the paper [18].

Recall the well-known string equation hτ0

n

Y

i=1

τ_kiig =

n

X

j=1

hτ_kj−1Y

i6=j

τ_kiig

and the dilaton equation hτ1

n

Y

i=1

τ_kiig = (2g−2 +n)h

n

Y

i=1

τ_kiig.

Note that the string equation can be equivalently written as F(x₁, . . . , x_n,0) =Xⁿ

j=1

x_j

F(x₁, . . . , x_n).

Proposition 2.3. Let n ≥ 2. We have the following recursive for- mula of n-point functions.

(2g+n−1)F_g(x₁, . . . , x_n) = P_n

j=1x_j3

12 F_g−1(x₁, . . . , x_n)

+ 1

2 P_n

j=1x_j

g

X

g^′=0

X

n=I‘ J

X

i∈I

x_i

!2

X

i∈J

x_i

!2

F_g^′(x_I)F_g−g^′(x_J).

Proof. From Proposition 2.2, we have G_g(x₁, . . . , x_n)

= X

r+s=g

(2r+n−3)!!

4^s(2g+n−1)!!P_r(x₁, . . . , x_n)∆(x₁, . . . , x_n)^s

= 1

2g+n−1P_g(x₁, . . . , x_n)

+ X

r+s=g−1

(2r+n−3)!!

4^s+1(2g+n−1)!!P_r(x₁, . . . , x_n)∆(x₁, . . . , x_n)^s+1

= 1

(2g+n−1)P_g(x₁, . . . , x_n) +∆(x₁, . . . , x_n)

4(2g+n−1)G_g−1(x₁, . . . , x_n).

We define H= exp

Pn i=1x³_i

24

, H⁻¹ = exp

−Pn i=1x³_i 24

,

H_d= 1 d!

Pn i=1x³_i

24 d

, H_d⁻¹ = 1 d!

−P_n

i=1x³_i 24

d

.

Note thatP_d

i=0H_iH_d−i⁻¹ = 0 if d >0.

Let LHS and RHS denote the left and right hand side of the recursion in the lemma. We have

H⁻¹·RHS=

∞

X

g=0



 1 12

n

X

i=1

xi

!3

Gg−1(x1, . . . , xn) +Pg(x1, . . . , xn)





=

∞

X

g=0

(2g+n−1)G_g(x₁, . . . , x_n) + 1 12

n

X

i=1

x³_i

!

G_g−1(x₁, . . . , x_n)

!

H⁻¹·LHS=

∞

X

g=0

X

a+b+c=g

(2a+ 2b+n−1)G_a(x₁, . . . , x_n)H_bH_c⁻¹

=

∞

X

g=0 g

X

a=0

(2a+n−1)G_a(x₁, . . . , x_n) X

b+c=g−a

H_bH_c⁻¹

+

∞

X

g=0

X

a+b+c=g

G_a(x₁, . . . , x_n)2bH_bH_c⁻¹

=

∞

X

g=0

(2g+n−1)G_g(x₁, . . . , x_n) +

∞

X

g=0

1 12

n

X

i=1

x³_i

!

G_g−1(x₁, . . . , x_n).

q.e.d.

Only very recently, we realize that Proposition 2.3 has already been embodied in the first KdV equation of the Witten-Kontsevich theorem.

The KdV hierarchy is the following hierarchy of differential equations forn≥1,

∂U

∂t_n = ∂

∂t₀R_n+1,

where R_n are Gelfand-Dikii differential polynomials in U, ∂U/∂t₀,

∂²U/∂t²₀, . . ., defined recursively by R1=U, ∂Rn+1

∂t₀ = 1

2n+ 1 ∂U

∂t₀Rn+ 2U∂Rn

∂t₀ + 1 4

∂³

∂t³₀Rn

. It is easy to see that

R₂ = 1

2U²+ 1 12

∂²U

∂t²₀ , R₃ = 1

6U³+ U 12

∂³U

∂t³₀ + 1 24

∂U

∂t₀ 2

+ 1 240

∂⁴U

∂t⁴₀ , ...

The Witten-Kontsevich theorem states that the generating function F(t₀, t₁, . . .) =X

g

X

n

h

∞

Y

i=0

τ_iⁿⁱig

∞

Y

i=0

tⁿ_iⁱ ni!

is a τ-function for the KdV hierarchy, i.e. ∂²F/∂t²₀ obeys all equations in the KdV hierarchy. The first equation in the KdV hierarchy is the classical KdV equation

∂U

∂t₁ =U∂U

∂t₀ + 1 12

∂³U

∂t³₀ .

By the Witten-Kontsevich theorem, we have

∂³F

∂t₁∂t²₀ = ∂F

∂t²₀

∂F

∂t³₀ + 1 12

∂⁵F

∂t⁵₀ .

Integrating each side with respect to t₀ and putting hhτ_k1· · ·τ_knii:=

∂ⁿF/∂t_k1· · ·∂t_kn, we get hhτ0τ1ii= 1

12hhτ₀⁴ii+1

2hhτ₀²iihhτ₀²ii.

Then Proposition 2.3 follows by applying the dilaton equation.

3. The Faber intersection number conjecture

Now we explain our approach to prove identity (3), hence the Faber intersection number conjecture. We establish its relationship with n- point functions.

For the sake of brevity, we introduce the following notations L^a,b_g (y, x1. . . , xn)

=

g

X

g^′=0

X

n=I‘ J

y+X

i∈I

xi

a

−y+X

i∈J

xi

b

F_g^′(y, xI)F_g−g^′(−y, xJ), where a, b ∈ Z. We regard L^a,bg (y, x₁. . . , x_n) as a formal series in Q[x₁, . . . , x_n][[y, y⁻¹]] with degy <∞.

We now prove that the Faber intersection number conjecture can be reduced to three statements about the coefficients of the above func- tions.

Proposition 3.1. We have i)

L^0,0_g (y, x1. . . , xn)

y^2g−2 = 0;

ii) For k >2g,

L^2,2_g (y, x₁. . . , x_n)

y^k = 0;

iii) For d_j ≥1 andPn

j=1d_j =g+n, L^2,2_g (y, x₁. . . , x_n)

y^2gQn

j=1x^dj_j = (2g+n+ 1)!

4^g(2g+ 1)!Q_n

j=1(2d_j−1)!!. In fact, Proposition 3.1 is a special case of more general results proved in the next section. Clearly identities (i) and (ii) of the following corol- lary add up to the desired identity (3).

Corollary 3.2. We have

i) Let d_j ≥0 and P_n

j=1d_j =g+n−2. Then h

n

Y

j=1

τ_djτ_2gig =

n

X

j=1

hτdj+2g−1

Y

i6=j

τ_diig

− 1 2

X

n=I‘ J

2g−2

X

j=0

(−1)^jhτj

Y

i∈I

τ_diig^′hτ2g−2−j

Y

i∈J

τ_diig−g^′; ii) Let dj ≥1 and Pn

j=1(dj−1) =g−1. Then

2g

X

j=0

(−1)^jhτ2g−jτ_j

n

Y

i=1

τ_diig= (2g+n−1)!

4^g(2g+ 1)!Q_n

j=1(2d_j−1)!!; iii) Let k > g, d_j ≥0 and P_n

j=1d_j = 3g+n−2k−2. Then

2k

X

j=0

(−1)^jhτ2k−jτ_j

n

Y

i=1

τ_diig = 0.

Proof. Since one and two-point functions in genus 0 are F₀(x) = 1

x², F₀(x, y) = 1 x+y =

∞

X

k=0

(−1)^k x^k y^k+1, it is consistent to define

hτ−2i0 = 1, hτ_kτ_−1−ki0 = (−1)^k, k ≥0.

By allowing the index to run over all integers, we have

1 2

X

n=I‘ J

2g−2

X

j=0

(−1)^jhτ_jY

i∈I

τ_dii_g^′hτ_2g−2−jY

i∈J

τ_dii_g−g^′

+h

n

Y

j=1

τ_djτ2gig−

n

X

j=1

hτ_dj+2g−1Y

i6=j

τ_diig

= 1 2

X

n=I‘ J

X

j∈Z

(−1)^jhτj

Y

i∈I

τ_dii_g^′hτ2g−2−j

Y

i∈J

τ_dii_g−g^′

=





g

X

g^′=0

X

n=I‘ J

F_g^′(y, x_I)F_g−g^′(−y, x_J)





y^2g−2Qn i=1x^di_i

=

L^0,0_g (y, x₁, . . . , x_n)

y^2g−2Qn

i=1x^di_i = 0.

From Proposition 2.3, we have 1

2





n

X

j=1

xj



Fg(x1, . . . , xn) = Pn

j=1x_j4

24(2g+n−1)Fg−1(x1, . . . , xn)

+ 1

2(2g+n−1)



L^2,2_g (y, x_n) +

g

X

g^′=0

X

n=I‘ J

X

i∈I

x_i

!2

X

i∈J

x_i

!2

×F_g^′(y,−y, xI)F_g−g^′(x_J)

! . By Proposition 3.1(ii)-(iii), we can use Proposition 2.3 to inductively prove

2k

X

j=0

(−1)^jhτ2k−jτ_j

n

Y

i=1

τ_diig= [F_g(y,−y, x1, . . . , x_n)]_y2k = 0, fork > g and we have

2g

X

j=0

(−1)^jhτ2g−jτ_jτ₀

n

Y

i=1

τ_diig = (2g+n)!

4^g(2g+ 1)!Q_n

j=1(2d_j−1)!!, which, from the string equation and induction on the maximum index (say d₁) among {di}, implies (by the dilaton equation, we may assume d_i≥2)

2g

X

j=0

(−1)^jhτ2g−jτj n

Y

i=1

τ_diig

=

2g

X

j=0

(−1)^jhτ0τ_2g−jτ_jτ_d1+1

n

Y

i=2

τ_diig

−

n

X

k=2 2g

X

j=0

(−1)^jhτ2g−jτ_jτ_d1+1τ_dk−1 Y

i6=1,k

τ_diig

= (2g+n)!

4^g(2g+ 1)!Qn

j=1(2d_j−1)!!(2d₁ + 1)

−

n

X

k=2

(2g+n−1)!(2d_k−1) 4^g(2g+ 1)!Q_n

j=1(2d_j −1)!!(2d₁ + 1)

= (2g+n−1)!

4^g(2g+ 1)!Q_n

j=1(2d_j−1)!!.

q.e.d.

So in order to prove the Faber intersection number conjecture, we only need to prove the three statements (i)-(iii) in Proposition 3.1 about n- point functions. Actually we will prove more general results which are stated as main theorems, Theorems 4.4 and 4.5 in the next section.

Proposition 3.1, therefore the Faber intersection number conjecture, is a special case of these theorems.

4. Proof of main theorems The binomial coefficients _k^p

, fork≥0, p∈Z are given by p

k

=







0, k <0,

1, k= 0,

p(p−1)···(p−k+1)

k! , k≥1.

Lemma 4.1. Let a, b∈Z andn≥0. Then

n

X

i=0

i+a i

n−i+b n−i

=

n+a+b+ 1 n

. Proof. Note that

p k

=

p−1 k

+

p−1 k−1

.

By denoting the left-hand side of the above equation by A_n(a, b), we have

A_n(a, b) =A_n(a−1, b) +A_n−1(a, b).

First we argue by induction on nand |b| to prove A_n(0, b) =

n+b+ 1 n

.

Then we argue by induction on nand |a|to prove A_n(a, b) =

n+a+b+ 1 n

.

q.e.d.

We now prove two lemmas that will serve as base cases for our in- ductive arguments.

Lemma 4.2. Let a, b∈Z andk≥2g−3 +a+b. Then i)

h

L^a,b_g (y, x)i

y^k = 0,

ii)

h

L^a,b_g (y, x)i

y^2g−4+a+bx^g+1 = (−1)^b(2g−2 +a+b) 4^g(2g+ 1)!! .

Proof. Here we recall the definition of normalizedn-point functions G(x₁, . . . , x_n) = exp −P_n

j=1x³_j 24

!

·F(x₁, . . . , x_n).

In particular, we have G(x) = 1

x², G(x, y) = 1 x+y

X

k≥0

k!

(2k+ 1)!

1

2xy(x+y) k

. By definition

X

g≥0

L^a,b_g (y, x₁. . . , x_n) =

exp Pn

j=1x³_j 24

! X

n=I‘ J

y+X

i∈I

x_ia

−y+X

i∈J

x_ib

G(y, x_I)G(−y, xJ), So for statements (i) and (ii), it is not difficult to see that we only need to prove that for k≥2g−3 +a+b,

h

y^a−2(−y+x)^bG_g(−y, x) + (−y)^b−2(y+x)^aG_g(y, x)i

y^k = 0 and

h

y^a−2(−y+x)^bG_g(−y, x) + (−y)^b−2(y+x)^aG_g(y, x)i

y^2g−4+a+bx^g+1

= (−1)^b(2g−2 +a+b) 4^g(2g+ 1)!! .

Both follow easily from the explicit formula ofG(y, x). q.e.d.

Lemma 4.3. Let a, b∈Z andk≥a+b−3. Then i)

hL^a,b₀ (y, x₁, . . . , x_n)i

y^k = 0, ii)

h

L^a,b₀ (y, x1, . . . , xn)i

y^a+b−4Qn j=1xj

= (−1)^b(a+b+n−3)!

(a+b−3)! . Proof. Since

F₀(x₁, . . . , x_n) = (x₁+· · ·+x_n)ⁿ⁻³, we have by definition

L^a,b₀ (y, x₁. . . , x_n) = X

n=I‘ J

y+X

i∈I

x_i|I|−2+a

−y+X

i∈J

x_i|J|−2+b

.

For any monomial y^kQn

j=1x^d_j^j in L^a,b₀ (y, x1. . . , xn), if k ≥ a+b−3, then there must be somed_j = 0. We may assumed_n= 0, then

L^a,b₀ (y, x₁. . . , x_n−1,0)

= X

{1,...,n−1}=I‘ J

y+X

i∈I

x_i|I|−1+a

−y+X

i∈J

x_i|J|−2+b

+

y+X

i∈I

x_i|I|−2+a

−y+X

i∈J

x_i|J|−1+b!

=





n−1

X

j=1

x_j





X

{1,...,n−1}=I‘ J

x₁+X

i∈I

x_i|I|−2+a

−x₁+X

i∈J

x_i|J|−2+b

=





n−1

X

j=1

x_j



L^a,b₀ (y, x₁. . . , x_n−1).

So (i) follows by induction onn. By applying Lemma 4.1 we have

hL^a,b₀ (y, x₁, . . . , x_n)i

y^a+b−4Qn j=1xj

= (−1)^b

n

X

|I|=0

|I| −2 +a

|I|

|I|!

|j| −2 +b

|J|

|J|!

n

|I|

= (−1)^bn!

n

X

i=0

i−2 +a i

n−i−2 +b n−i

= (−1)^bn!

a+b+n−3 n

= (−1)^b(a+b+n−3)!

(a+b−3)! .

So we have proved (ii). q.e.d.

Theorem 4.4. Let a, b∈Z and k≥2g−3 +a+b. Then h

L^a,b_g (y, x1. . . , xn)i

y^k = 0.

Proof. We will argue by induction ongandn, since the theorem holds forg= 0 or n= 1 as proved in the above lemmas. We have

(2g+n)L^a,b_g (y, x₁. . . , x_n)

=

g

X

g^′=0

X

n=I‘ J

y+X

i∈I

x_ia

−y+X

i∈J

x_ib

(2g^′+|I|)Fg^′(y, x_I)F_g−g^′(−y, xJ) +

g

X

g^′=0

X

n=I‘ J

y+X

i∈I

x_ia

−y+X

i∈J

x_ib

F_g^′(y, x_I)(2g−2g^′+|J|)F_g−g^′(−y, xJ).

SubstitutingF_g^′(y, x_I) by Propostion 2.3,





g

X

g^′=0

X

n=I‘ J

y+X

i∈I

x_ia

−y+X

i∈J

x_ib

(2g^′ +|I|)

×F_g^′(y, x_I)F_g−g^′(−y, x_J)

#

y^k

= 1 12

hL^a+3,b_g−1 (y, x₁, . . . , x_n)i

y^k

+





g

X

g^′=0

X

s≥0

a−1 s

X

n=I‘ J

F_g^′(x_I) X

i∈I

x_is+2

L^a+1−s,b_g−g′ (y, x_J)





y^k

.

Note that in the last term of the above equation, |J| < n. So by induction, fork≥2g−3 +a+b, the sums vanish except for g^′ = 0 and s= 0, namely the term



 X

n=I‘ J

X

i∈I

xi

|I|−1

L^a+1,b_g (y, xJ)





y^k

.

Letd_j ≥1 for 1≤j≤n. By induction, it is not difficult to see from the above that

(2g+n)h

L^a,b_g (y, x₁. . . , x_n)i

y^kQn j=1x^dj_j

= 1 12

h

L^a+3,b_g−1 (y, x₁, . . . , x_n) +L^a,b+3_g−1 (y, x₁, . . . , x_n)i

y^kQn j=1x^dj_j .

By induction, we have

0 =





n

X

j=1

x_j



 h

L^a+1,b+1_g−1 (y, x₁, . . . , x_n)i

y^k fork≥2g−3 +a+b

=h

L^a+2,b+1_g−1 (y, x₁, . . . , x_n) +L^a+1,b+2_g−1 (y, x₁, . . . , x_n)i

y^k

and

0 =





n

X

j=1

x_j





3

h

L^a,b_g−1(y, x₁, . . . , x_n)i

y^k fork≥2g−5 +a+b

=h

L^a+3,b_g−1 (y, x₁, . . . , x_n) +L^a,b+3_g−1 (y, x₁, . . . , x_n)i

y^k

+ 3h

L^a+2,b+1_g−1 (y, x₁, . . . , x_n) +L^a+1,b+2_g−1 (y, x₁, . . . , x_n)i

y^k

=h

L^a+3,b_g−1 (y, x₁, . . . , x_n) +L^a,b+3_g−1 (y, x₁, . . . , x_n)i

y^k.

So we have proved that h

L^a,b_g (y, x₁. . . , x_n)i

y^kQn

j=1x^dj_j = 0, ford_j ≥1.

If some d_j is zero, the above identity still holds by applying the string equation

L^a,b_g (y, x₁. . . , x_n,0) =





n

X

j=1

x_j



L^a,b_g (y, x₁. . . , x_n).

So we proved the theorem. q.e.d.

Theorem 4.5. Let a, b∈Z, d_j ≥1 and P

jd_j =g+n. Then h

L^a,b_g (y, x₁. . . , x_n)i

y^2g−4+a+bQn j=1x^dj_j

= (−1)^b(2g−3 +n+a+b)!

4^g(2g−3 +a+b)!Q_n

j=1(2d_j−1)!!.

Proof. By the dilaton equation, we may assume d_j ≥ 2. As in the proof of the above theorem, we have

(2g+n)h

L^a,b_g (y, x₁. . . , x_n)i

y^2g−4+a+bQn j=1x^dj_j

= 1 12

hL^a+3,b_g−1 (y, x_n) +L^a,b+3_g−1 (y, x_n)i

y^2g−4+a+bQn j=1x^dj_j

=−1 4

h

L^a+2,b+1_g−1 (y, xn) +L^a+1,b+2_g−1 (y, xn)i

y^2g−4+a+bQn j=1x^dj_j

=−1 4

" _n X

i=1

xi

!

L^a+1,b+1_g−1 (y, xn)

#

y^2g−4+a+bQn j=1x^dj_j

=−1 4

n

X

j=1

h

L^a+1,b+1_g−1 (y, x_n)i

y^2g−4+a+bx^dj_j ⁻¹Q

i6=jx^di_i

= (−1)^b(2g−3 +n+a+b)!

4^g(2g−3 +a+b)!Qn

j=1(2d_j−1)!!

n

X

j=1

(2d_j−1)

= (2g+n) (−1)^b(2g−3 +n+a+b)!

4^g(2g−3 +a+b)!Qn

j=1(2dj−1)!!.

So we have proved the theorem. q.e.d.

All the three statements in Proposition 3.1 are particular cases of Theorems 2.4 and 2.5. We thus conclude the proof of the Faber inter- section number conjecture.

The following corollaries ware stated as conjectures in our previous paper [17].

Corollary 4.6. Let d_j ≥1 and P_n

j=1(d_j−1) =g. Then (2g−3 +n)!

2^2g+1(2g−3)!Qn

j=1(2d_j−1)!!

=hτ2g−2 n

Y

j=1

τ_djig−

n

X

j=1

hτdj+2g−3

Y

i6=j

τ_diig

+ 1 2

X

n=I‘ J

2g−4

X

j=0

(−1)^jhτj

Y

i∈I

τ_diig^′hτ2g−4−j

Y

i∈J

τ_diig−g^′.

Proof. Since the right hand side is just 1

2

L^0,0_g (y, x₁. . . , x_n)

y^2g−4Qn j=1x^dj_j ,

the result follows from Theorem 4.5. q.e.d.

Corollary 4.7. Let g≥2, d_j ≥1 and P_n

j=1(d_j−1) =g. Then

−(2g−2)!

|B2g−2| Z

Mg,n

ψ^d₁¹· · ·ψ^d_nⁿch_2g−3(E)

= 2g−2

|B2g−2| Z

Mg,n

ψ^d₁¹· · ·ψ^d_nⁿλ_g−1λ_g−2−3 Z

Mg,n

ψ₁^d1· · ·ψ^d_nⁿλ_g−3λ_g

!

= 1 2

2g−4

X

j=0

(−1)^jhτ2g−4−jτ_jτ_d1· · ·τ_dnig−1

+ (2g−3 +n)!

2^2g+1(2g−3)!· 1 Q_n

j=1(2d_j −1)!!. Proof. We apply Mumford’s formulae [22]

(2g−3)!·ch2g−3(E) = (−1)^g−1(3λg−3λg−λg−1λg−2),

ch_2g−3(E) = B_2g−2 (2g−2)!

"

κ_2g−3−

n

X

i=1

ψ^2g−3_i

+1 2

X

ξ∈∆

l_ξ∗

2g−4

X

i=0

ψⁱ_n+1(−ψn+2)^2g−4−i

! # .

So the identity follows from Corollary 4.6. q.e.d.

Both Theorems 4.4 and 4.5 can be extended without difficulty.

Let us use the notation

L_g(y, z_a, w_b, x_n)

=

g

X

g^′=0

X

n=I‘ J

F_g^′(y, z₁, . . . , z_a, x_I)F_g−g^′(−y, w1, . . . , w_b, x_J).

Theorem 4.8. Let a≥0, b≥0, n≥1. We have i) For k≥2g−3 +a+b,

Lg(y, za, w_b, xn)

y^k = 0.

ii) For r_j ≥0, s_j ≥0, d_j ≥1 and P

r_j+P

s_j+P

d_j =g+n, L_g(y, z_a, w_b, x_n)

y^2g−4+a+bQa

j=1z_j^rjQb

j=1w^sj_j Qn j=1x^dj_j

= 1

Q_a

j=1(2r_j+ 1)!!Q_b

j=1(2s_j+ 1)!!

· (−1)^b(2g−3 +n+a+b)!

4^g(2g−3 +a+b)!Q_n

j=1(2d_j −1)!!.

iii) For r_j ≥0, s_j ≥0, d_j ≥1, P

r_j+P

s_j +P

d_j =g+n+ 1 and u,#{rj = 0}, v,#{sj = 0}, w,#{dj = 1},

L_g(y, z_a, w_b, x_n)

y^2g−5+a+bQa

j=1z_j^rjQb

j=1w^sj_j Qn

j=1x^dj_j = C

Q_a

j=1(2r_j+ 1)!!Q_b

j=1(2s_j+ 1)!!· (−1)^b(2g−3 +n+a+b)!

4^g(2g−4 +a+b)!Q_n

j=1(2d_j −1)!!, where the constant C is given by

C,

a

X

j=1

rj−

b

X

j=1

sj+a−b

2 + (5−u)u−(5−v)v 2(2g+n+a+b−3−w). Proof. Wheng= 0, the proof is an easy verification. Let p, q∈Z.

L^p,q_g (y, za, w_b, xn)

=

g

X

g^′=0

X

n=I‘ J

(y+

a

X

i=1

z_i+X

i∈I

x_i)^p(−y+

b

X

i=1

w_i+X

i∈J

x_i)^q

×F_g^′(y, z₁, . . . , z_a, x_I)F_g−g^′(−y, w1, . . . , w_b, x_J).

Exactly the same argument of Theorem 4.4 will prove that for k ≥ 2g−3 +p+q+a+b,

[L^p,q_g (y, z_a, w_b, x_n)]_yk = 0.

Statements (ii) and (iii) can also be proved similarly as Theorem 4.5.

q.e.d.

Theorem 4.8 proves all conjectures in Section 3 of [17]. We may write down the coefficients of L_g(y, z_a, w_b, x_n) explicitly to get a lot of interesting identities of intersection numbers. For example, when a= 1, b= 0,

Lg(y, z, xn)

y^kz^rQn j=1x^dj_j

= X

n=I‘ J

k

X

j=0

(−1)^jhτj

Y

i∈I

τ_diig^′hτk−jτ_rY

i∈J

τ_diig−g^′

+hτk+2τ_r

n

Y

j=1

τ_djig−(−1)^khτk+r+1 n

Y

j=1

τ_djig−

n

X

j=1

hτrτ_dj+k+1Y

i6=j

τ_diig.